Mrs May reportedly told him that EU rules would oblige Ireland to construct border infrastructure to protect the single market and warned against any move that might anger moderate nationalists in Northern Ireland.

Mrs May “slapped him down very hard” according to one MP present at the meeting told the Times.

Despite the reports, a Brexiteer source said it was “tosh” to suggest that the Union could be jeopardised and suggested that the reported “slap-down” was “Number 10-ese for “they disagreed, she said something weird and implausible”.

The PM’s Brexit war cabinet is due to meet again as discussions over how to resolve the deadlock on the customs union continue.

Theresa May clashed with Jacob Rees-Mogg during a briefing at Downing Street

Members of two ministerial working groups spent Monday discussing possible tweaks to the “customs partnership” and “maximum facilitation” plans that have divided the Cabinet, while the Prime Minister herself briefed Tory MPs on details of the two options.

The EU is pressing for “substantive progress” from Britain on the future of the Irish border in time for a crunch European Council summit just six weeks away.

But the meeting of the Brexit sub-committee is thought unlikely to unite behind a solution when it meets on Tuesday.

Tory Rees-Mogg, who leads the European Research Group of Conservative Brexiteers, warned the Prime Minister she must show more “firmness and tenacity”.

Writing in the Daily Telegraph, the MP accused Brussels of treating Britain with “disdain” and using the Irish border as a way to stop Brexit.

“If Brussels or Dublin insist upon rejecting all the practical approaches that we propose it is, with regret, that we will have to graciously accept their rejection,” he added.

“If they either do not want to or simply cannot broker a deal, a deal will not be done.”

Under a customs partnership, the UK would collect tariffs for Brussels. The alternative model being considered by the Cabinet is known as maximum facilitation, or “Max Fac”, which would rely on new technology and trusted trader schemes.

Michel Barnier, the EU’s chief Brexit negotiator, told ministers from the remaining 27 member states that “little progress” had been made on the issues of Ireland and governance since the Council last met in March.

Yesterday, former Lib Dem leader Nick Clegg said EU leaders were “baffled” about what the UK wants out of Brexit.